Natural bundle

In differential geometry, a field in mathematics, a natural bundle is any fiber bundle associated to the higher order frame bundle $F^{r}(M)$ , for some $r\geq 1$ . In other words, its transition functions depend functionally on local changes of coordinates in the base manifold $M$ together with their partial derivatives up to order at most $r$ .^[1]^[2]

The concept of a natural bundle was introduced in 1972 by Albert Nijenhuis as a modern reformulation of the classical concept of an arbitrary bundle of geometric objects.^[3]

Definition

Let ${\mathcal {M}}f$ denote the category of smooth manifolds and smooth maps and ${\mathcal {M}}f_{n}$ the category of smooth $n$ -dimensional manifolds and local diffeomorphisms. Consider also the category ${\mathcal {FM}}$ of fibred manifolds and bundle morphisms, and the functor $B:{\mathcal {FM}}\to {\mathcal {M}}f$ associating to any fibred manifold its base manifold.

A natural bundle (or bundle functor) is a functor $F:{\mathcal {M}}f_{n}\to {\mathcal {FM}}$ satisfying the following three properties:

$B\circ F=\mathrm {id}$ , i.e. $F(M)$ is a fibred manifold over $M$ , with projection denoted by $p_{M}:F(M)\to M$ ;
if $U\subseteq M$ is an open submanifold, with inclusion map $i:U\hookrightarrow M$ , then $F(U)$ coincides with $p_{M}^{-1}(U)\subseteq F(M)$ , and $F(i):F(U)\to F(M)$ is the inclusion $p^{-1}(U)\hookrightarrow F(M)$ ;
for any smooth map $f:P\times M\to N$ such that $f(p,\cdot ):M\to N$ is a local diffeomorphism for every $p\in P$ , then the function $P\times F(M)\to F(N),(p,x)\mapsto F(f(p,\cdot ))(x)$ is smooth.

As a consequence of the first condition, one has a natural transformation $p:F\to \mathrm {id} _{{\mathcal {M}}f_{n}}$ .

Finite order natural bundles

A natural bundle $F:{\mathcal {M}}f_{n}\to {\mathcal {FM}}$ is called of finite order $r$ if, for every local diffeomorphism $f:M\to N$ and every point $x\in M$ , the map $F(f)_{x}:F(M)_{x}\to F(N)_{f(x)}$ depends only on the jet $j_{x}^{r}f$ . Equivalently, for every local diffeomorphisms $f,g:M\to N$ and every point $x\in M$ , one has $j_{x}^{r}f=j_{x}^{r}g\Rightarrow F(f)|_{F(M)_{x}}=F(g)|_{F(M)_{x}}.$ Natural bundles of order $r$ coincide with the associated fibre bundles to the $r$ -th order frame bundles $F^{r}(M)$ .

After various intermediate cases,^[1]^[4] it was proved by Epstein and Thurston that all natural bundles have finite order.^[2]

Natural $\Gamma$ -bundles

The notion of natural $\Gamma$ -bundle arises from that of natural bundle by restricting to the suitable categories of $\Gamma$ -manifolds and of $\Gamma$ -fibred manifolds, where $\Gamma$ is a pseudogroup. The case when $\Gamma$ is the pseudogroup of all diffeomorphisms between open subsets of $\mathbb {R} ^{n}$ recovers the ordinary notion of natural bundle.

Under suitable assumptions, natural $\Gamma$ -bundles have finite order as well.^[5]^[6]^[7]

Examples

An example of natural bundle (of first order) is the tangent bundle $TM$ of a manifold $M$ .

Other examples include the cotangent bundles, the bundles of metrics of signature $(r,s)$ and the bundle of linear connections.^[8]

Notes

^ ^a ^b Palais, Richard S.; Terng, Chuu-Lian (1977-01-01). "Natural bundles have finite order". Topology. 16 (3): 271–277. doi:10.1016/0040-9383(77)90008-8. ISSN 0040-9383.
^ ^a ^b Epstein, D. B. A.; Thurston, W. P. (1979). "Transformation Groups and Natural Bundles". Proceedings of the London Mathematical Society. s3-38 (2): 219–236. doi:10.1112/plms/s3-38.2.219.
^ Albert, Nijenhuis (1972). "Natural bundles and their general properties" (PDF). Differential Geometry (in honor of Kentaro Yano). Tokyo: Kinokuniya: 317–334.
^ Terng, Chuu Lian (1978). "Natural Vector Bundles and Natural Differential Operators". American Journal of Mathematics. 100 (4): 775–828. doi:10.2307/2373910. ISSN 0002-9327.
^ Slovák, Jan (1991). "Bundle functors on fibred manifolds". Annals of Global Analysis and Geometry. 9 (2): 129–143. doi:10.1007/BF00776852. ISSN 0232-704X.
^ Kolář, Ivan; Slovák, Jan; Michor, Peter W. (1993). Natural Operations in Differential Geometry. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-662-02950-3. ISBN 978-3-642-08149-1.
^ Benalili, Mohamed (1994-09-01). "Fibrés naturels sur la catégorie des Γ-variétés" [Natural bundles on the category of Γ-manifolds]. Rendiconti del Circolo Matematico di Palermo Series 2 (in French). 43 (3): 309–328. doi:10.1007/BF02844245. ISSN 1973-4409.
^ Fatibene, Lorenzo; Francaviglia, Mauro (2003). Natural and Gauge Natural Formalism for Classical Field Theorie. Springer. doi:10.1007/978-94-017-2384-8. ISBN 978-1-4020-1703-2.

References

Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag, archived from the original (PDF) on 2017-03-30, retrieved 2017-08-15
Krupka, Demeter; Janyška, Josef (1990), Lectures on differential invariants, Univerzita J. E. Purkyně V Brně, ISBN 80-210-0165-8
Saunders, D.J. (1989), The geometry of jet bundles, Cambridge University Press, ISBN 0-521-36948-7

[:0-1] Palais, Richard S.; Terng, Chuu-Lian (1977-01-01). "Natural bundles have finite order". Topology. 16 (3): 271–277. doi:10.1016/0040-9383(77)90008-8. ISSN 0040-9383.

[:1-2] Epstein, D. B. A.; Thurston, W. P. (1979). "Transformation Groups and Natural Bundles". Proceedings of the London Mathematical Society. s3-38 (2): 219–236. doi:10.1112/plms/s3-38.2.219.

[3] Albert, Nijenhuis (1972). "Natural bundles and their general properties" (PDF). Differential Geometry (in honor of Kentaro Yano). Tokyo: Kinokuniya: 317–334.

[4] Terng, Chuu Lian (1978). "Natural Vector Bundles and Natural Differential Operators". American Journal of Mathematics. 100 (4): 775–828. doi:10.2307/2373910. ISSN 0002-9327.

[5] Slovák, Jan (1991). "Bundle functors on fibred manifolds". Annals of Global Analysis and Geometry. 9 (2): 129–143. doi:10.1007/BF00776852. ISSN 0232-704X.

[6] Kolář, Ivan; Slovák, Jan; Michor, Peter W. (1993). Natural Operations in Differential Geometry. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-662-02950-3. ISBN 978-3-642-08149-1.

[7] Benalili, Mohamed (1994-09-01). "Fibrés naturels sur la catégorie des Γ-variétés" [Natural bundles on the category of Γ-manifolds]. Rendiconti del Circolo Matematico di Palermo Series 2 (in French). 43 (3): 309–328. doi:10.1007/BF02844245. ISSN 1973-4409.

[8] Fatibene, Lorenzo; Francaviglia, Mauro (2003). Natural and Gauge Natural Formalism for Classical Field Theorie. Springer. doi:10.1007/978-94-017-2384-8. ISBN 978-1-4020-1703-2.

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